Probability

Brain teasers and puzzles involving randomness, probability, and statistics

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Brain Teasers and Puzzles Learning

The Monty Hall Problem Explained

The Monty Hall Problem

You are on a game show in which there are three identical doors, one with a car behind it and two with goats behind them. You must pick one door, and you win if that door has the car behind it.

After you pick a door, the host of the game show always opens a door you didn’t choose that has a goat behind it. This leaves the door you chose and one other remaining door, and you are given the option to switch your choice to the other remaining door.

Should you switch or should you stick to your original choice? What chance of winning would that give you?

The History

The Monty Hall Problem is a classic probability puzzle, named for its similarity to the game show “Let’s Make a Deal”, which was hosted by Monty Hall. The problem was made famous when Marilyn vos Savant answered it correctly in her column in a popular magazine, and thousands of readers wrote letters to the magazine arguing her solution was wrong!

The solution can be counter-intuitive, so give it some thought and then scroll down to see the Monty Hall Problem explained.

More info and solution “The Monty Hall Problem Explained”
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Brain Teasers and Puzzles

Same Birthday in Line

There is a long line of people waiting to see a new movie. They announce that the first person to have the same birthday as someone standing before them in the line gets to meet one of the actors in the movie.

What place in line would maximize your chances of winning? Assume birthdays are uniformly distributed through the year.

More info and solution “Same Birthday in Line”
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Brain Teasers and Puzzles

Russian Roulette Riddle

In the morbid game of Russian Roulette, a partially loaded revolver with a six-chamber cylinder is randomly spun, pointed at one of the players, and fired. If the revolver landed on an empty chamber, the lucky player is safe, and the process is repeated with the next player. The obvious objective of the game is to not get shot.

You find yourself stuck in a game of Russian Roulette. A freshly loaded revolver is aimed at the first player, and it turns out to be an empty chamber. Your turn is next, and you are given the choice to either:

  • Spin the cylinder before pulling the trigger (i.e., you get a random new chamber)
  • Or just pull the trigger (i.e., let the revolver fire whatever is in the next chamber)

Which choice should you pick if the revolver was originally:

  1. Loaded with one bullet?
  2. Loaded with bullets in two random chambers?
  3. Loaded with bullets in two consecutive chambers?

Assume the revolver cannot misfire, and that spinning the cylinder lands on all chambers with equal probability.

Some variation of this Russian Roulette riddle was once asked in interviews at Jane Street, Susquehanna International Group (SIG), Facebook (now Meta), UBS, Capital One, and more.

More info and solution “Russian Roulette Riddle”
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Brain Teasers and Puzzles

Truel

A truel is a three-way duel. The three participants take turns firing one shot at whichever opponent they choose, until only one is remaining.

Allison, Ben, and Chase are in a truel, and have varying degrees of accuracy: Allison has a 50% chance of hitting her intended target, Ben has a 80% chance, and Chase has a 100% chance. Allison gets to shoot first, then Ben, then Chase, and repeating in that order until only one person is remaining.

Assume that the accuracy of all participants are publicly known, and everyone is trying to maximize their chances of winning. What is Allison’s optimal strategy, and what is her likelihood of winning under that strategy?

More info and solution “Truel”
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Brain Teasers and Puzzles

Reroll the Die

Suppose there is a game in which you roll a fair, 6-sided die and win dollars equal to the outcome of the roll. How much would you expect to win on average?

Suppose, if you don’t like the outcome of the roll, you can reroll the die once, and win dollars equal to the outcome of the 2nd roll (once you choose to reroll, you can no longer go back to the 1st roll). How much would you expect to win on average?

Suppose, if you don’t like the outcome of the 2nd roll, you can reroll the die once more, and win dollars equal to the outcome of the 3rd roll (once you choose to reroll, you can no longer go back to previous rolls). How much would you expect to win on average?

This was an actual brain teaser question once asked at Jane Street for an interview for an intern role.

More info and solution “Reroll the Die”
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Brain Teasers and Puzzles

How Many Daughters

A mother randomly selected two of her children to pick up a package from the post office. There was a 50% chance both children were daughters. How many daughters does this mother most likely have?

More info and solution “How Many Daughters”
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Brain Teasers and Puzzles

Tennis in 2 or 3 sets

In a best of 3 tennis match, the player that first wins 2 sets wins the match. For a 3-set tennis match, would you bet on it finishing in 2 or 3 sets?

This question was asked in an actual D. E. Shaw quant trading interview.

Preparing for a brain teaser interview? Check out our ultimate guide to brain teaser interviews.

More info and solution “Tennis in 2 or 3 sets”
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Brain Teasers and Puzzles

Vacant Room Probability

Your workplace has a phone room for employees to quietly make personal calls. The room has no windows, just a sign that can be switched from “Vacant” to “Occupied”. However, employees differ in how consistently they use the sign:

  • 1/2 of them always switch to “Occupied” when they enter and “Vacant” when they exit.
  • 1/4 of them ignore the sign altogether – the sign will always read the same before, during, and after their visit.
  • 1/4 of them always switching to “Occupied” when they enter, but always forget to switch back to “Vacant” when they exit.

If the room is actually occupied exactly 1/2 of the time, what is the probability the room is actually vacant when the sign reads “Vacant”?

More info and solution “Vacant Room Probability”
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Brain Teasers and Puzzles

Random Turns

You are in a city with streets on a perfect grid – every street is north-south or east-west. You are are driving north, and decide to randomly turn left or right with equal probability at the next 10 intersections.

After these 10 random turns, what is the probability you are still driving north?

More info and solution “Random Turns”
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Brain Teasers and Puzzles

Tale of Two Trains

There are two trains that run between two cities. The trains are identical and run on identical routes, so passengers have no preference between the two and would take whichever train that pulls into the station. The trains run at the same frequency: exactly once an hour.

You often travel between the two cities on a whim, and when you do so, you show up at the station at a completely random time. Yet after many trips over the years, you notice that you have taken one of the trains three times as often as the other. Is this just really bad/good luck, or is there another likely explanation?

More info and solution “Tale of Two Trains”
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Brain Teasers and Puzzles

Wrong Seat Probability

There is a fully-booked flight with 100 seats. The first person decides to ignore the seat assignments, and sits in a random seat. Each subsequent person sits in their assigned seats if available, or sits in a random unoccupied seat if not.

What is the probability that the last person happens to find their assigned seat unoccupied?

More info and solution “Wrong Seat Probability”
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Brain Teasers and Puzzles

Chance of Rain

You are driving to Seattle to meet some friends, and want to know whether you should bring an umbrella. You call up 3 of your friends who live there and independently ask them if it’s raining. Your friends like to mess with you, so each of them has a 1/3 chance of lying. If all 3 friends tell you it is raining, what is the probability it is actually raining there?

This question was asked in an actual Facebook data scientist/data analytics interview.

Preparing for a brain teaser interview? Check out our ultimate guide to brain teaser interviews.

More info and solution “Chance of Rain”
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Brain Teasers and Puzzles

Game the Coin Flip Game

A small company is losing money and the boss is looking for ways to cut costs. The boss is a foolish gambler, so he offers two employees the choice of either taking flat 10% pay cut, or playing the following coin flip game for each paycheck:

  1. The employees each flip a fair coin – 50% heads and 50% tails; the boss will ensure they are not pulling any tricks.
  2. They can see their own outcome but not the outcome of the other employee’s coin flip. They must then guess the outcome of the other employee’s coin flip.
  3. If at least one of them guesses correctly, they get their full paycheck.
  4. If both of them guess incorrectly, they get nothing.

The two employees take some time to work out a strategy, and then confidently accept the offer to play the coin flip game.

What strategy did they come up with that made them confident the coin flip game will give them more money?

More info and solution “Game the Coin Flip Game”